**1. Introduction**

66 Recent Advances in Crystallography

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In the field of X-ray crystal structure analysis, while the absolute values of structure factors are directly observed, phase information is lost in general. However, this problem (phase problem) has been overcome mainly by the direct method developed by Hauptmann and Karle except for protein crystals. In the case of protein crystal structure analysis, the isomorphous replacement method and/or anomalous dispersion method are mainly used to solve the phase problem. Phasing the structure factors is sometimes the most difficult process in protein crystallography.

On the other hand, It has been recognized for many years since the suggestion by Lipscomb in 1949 [12] that the phase information can be physically extracted, at least in principle, from X-ray diffraction profiles of three-beam cases in which transmitted and two reflected beams are simultaneously strong in the crystal. This suggestion was verified by Colella [3] that stimulated many authors [2, 4, 5, 21, 22] and let them investigate the multiple-beam (*n*-beam) method to solve the phase problem in protein crystallography.

The most primitive *n*-beam diffraction is the cases *n* = 3. The shape of three-beam rocking curve simply depends on the triplet phase invariant. In the case of protein crystallography, however, it is extremely difficult to realize such three-beam cases that transmitted and only two reflected beams are strong in the crystal, which is due to the extremely high density of reciprocal lattice nodes owing to the large size of unit cell of the crystal. Therefore, X-ray *n*-beam dynamical diffraction theory is necessary to solve the phase problem in protein crystallography. The Ewald-Laue (E-L) dynamical diffraction theory [7, 11] was extended to the three-beam cases in the late 1960's [8–10]. The numerical method to solve the *n*-beam (*n* ≥ 3) E-L theory was given by Colella [3]. Colella's method [3] to solve the *n*-beam E-L theory is applicable only to the case of crystals with planar surfaces.

©2012 Okitsu et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Okitsu et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

On the other hand, Okitsu and his coauthors [13, 14, 16, 17] extended the Takagi-Taupin (T-T) dynamical diffraction theory [18–20] to *n*-beam cases (*n* ∈ {3, 4, 5, 6, 8, 12}) and presented a numerical method to solve the theory. They showed six-beam pinhole topographs experimentally obtained and computer-simulated based on the *n*-beam T-T equation, between which excellent agreements were found. In reference [16], it was shown that the *n*-beam T-T equation can deal with X-ray wave field in an arbitrary-shaped crystal.

In the *n*-beam method to solve the phase problem in the protein crystal structure analysis, one of the difficulty is the shape of the crystal which is complex in general. Then, the above advantage of *n*-beam T-T equation over the E-L dynamical theory is important. The present authors have derived an *n*-beam T-T equation applicable for arbitrary number of *n*, which will be published elsewhere.

The *n*-beam T-T equation was derived in references [13, 17] from Takagi's fundamental equation of the dynamical theory [19]. In section 2 of the present chapter, however, the *n*-beam E-L theory is described at first. The *n*-beam T-T equation is derived by Fourier-transforming the *n*-beam E-L dynamical theory. Then, it is also described that the E-L theory can be derived from the T-T equation. This reveals a simple relation between the E-L and T-T formulations of X-ray dynamical diffraction theory. This equivalence between the E-L and T-T formulations has been implicitly recognized for many years but is explicitly described for the first time. In section 5, experimentally obtained and computer-simulated pinhole topographs are shown for *n* ∈ {3, 4, 5, 6, 8, 12}, which verifies the theory and the computer algorithm to solve it.

#### **2. Derivation of the** *n***-beam Takagi-Taupin equation**

#### **2.1. Description of the** *n***-beam Ewald-Laue dynamical diffraction theory**

The fundamental equation with the E-L formulation is given by [1, 2]

$$\frac{k\_i^2 - K^2}{k\_i^2} \mathcal{D}\_i = \sum \chi\_{h\_l - h\_l} \left[ \mathcal{D}\_j \right]\_{\perp, \mathbf{k}\_l}. \tag{1}$$

Here, *ki* is wavenumber of the *i*th numbered Bloch wave whose wave vector is **k**<sup>0</sup> + **h***<sup>i</sup>* where **k**<sup>0</sup> is the wave vector of the forward-diffracted wave in the crystal, *K*(= 1/*λ*) is the wavenumber of X-rays in vacuum, *D<sup>i</sup>* and *D<sup>j</sup>* are complex amplitude vectors of the *i*th and *j*th numbered Bloch waves, ∑ is an infinite summation for all combinations of *i* and *j*, *<sup>χ</sup>hi*−*hj* is Fourier coefficient of electric susceptibility and [*Dj*]⊥**k***<sup>i</sup>* is component vector of *D<sup>j</sup>* perpendicular to **k***i*, respectively.

By applying an approximation that *ki* + *K* ≈ 2*ki* to (1), the following equation is obtained,

$$\mathfrak{F}\_{i}\mathfrak{D}\_{i} = \frac{K}{2} \sum \chi\_{h\_{i}-h\_{j}} \left[ \mathfrak{D}\_{j} \right]\_{\perp \mathbf{k}\_{i}} \, \, \, \, \tag{2}$$
 
$$\text{where } \mathfrak{F}\_{i} = k\_{i} - \mathbf{K}.$$

Let the electric displacement vector *D<sup>i</sup>* be represented by a linear combination of scalar amplitudes as follows:

$$\mathcal{D}\_i = \mathcal{D}\_i^{(0)} \mathbf{e}\_i^{(0)} + \mathcal{D}\_i^{(1)} \mathbf{e}\_i^{(1)}.$$

Here, **e** (0) *<sup>i</sup>* and **e** (1) *<sup>i</sup>* are unit vectors perpendicular to **s***i*, where **s***<sup>i</sup>* is a unit vector parallel to **k***i*. **s***i*, **e** (0) *<sup>i</sup>* and **e** (1) *<sup>i</sup>* construct a right-handed orthogonal system in this order. (2) can be described as follows:

$$\begin{aligned} \xi \cos \Theta\_{\mathcal{B}} \mathcal{D}\_i^{(l)} &= -K \left( \mathcal{S}\_{i,0}^{(0)} \mathcal{\beta}^{(0)} + \mathcal{S}\_{i,0}^{(1)} \mathcal{\beta}^{(1)} \right) \mathcal{D}\_i^{(l)} \\ &+ \frac{K}{2} \sum\_{j=0}^{n-1} \sum\_{m=0}^1 \mathcal{C}\_{i,j}^{(l,m)} \chi\_{h\_i \cdots h\_j} \mathcal{D}\_j^{(m)}. \end{aligned} \tag{3}$$

where, *i*, *j* ∈ {0, 1, ··· , *n* − 1}, *n* ∈ {3, 4, 5, 6, 8, 12},

*l*, *m* ∈ {0, 1}.

Here, *S* and *C* are polarization factors defined by

2 Will-be-set-by-IN-TECH

On the other hand, Okitsu and his coauthors [13, 14, 16, 17] extended the Takagi-Taupin (T-T) dynamical diffraction theory [18–20] to *n*-beam cases (*n* ∈ {3, 4, 5, 6, 8, 12}) and presented a numerical method to solve the theory. They showed six-beam pinhole topographs experimentally obtained and computer-simulated based on the *n*-beam T-T equation, between which excellent agreements were found. In reference [16], it was shown that the *n*-beam T-T

In the *n*-beam method to solve the phase problem in the protein crystal structure analysis, one of the difficulty is the shape of the crystal which is complex in general. Then, the above advantage of *n*-beam T-T equation over the E-L dynamical theory is important. The present authors have derived an *n*-beam T-T equation applicable for arbitrary number of *n*, which will

The *n*-beam T-T equation was derived in references [13, 17] from Takagi's fundamental equation of the dynamical theory [19]. In section 2 of the present chapter, however, the *n*-beam E-L theory is described at first. The *n*-beam T-T equation is derived by Fourier-transforming the *n*-beam E-L dynamical theory. Then, it is also described that the E-L theory can be derived from the T-T equation. This reveals a simple relation between the E-L and T-T formulations of X-ray dynamical diffraction theory. This equivalence between the E-L and T-T formulations has been implicitly recognized for many years but is explicitly described for the first time. In section 5, experimentally obtained and computer-simulated pinhole topographs are shown for *n* ∈ {3, 4, 5, 6, 8, 12}, which verifies the theory and the computer algorithm to solve it.

equation can deal with X-ray wave field in an arbitrary-shaped crystal.

**2. Derivation of the** *n***-beam Takagi-Taupin equation**

*k*2 *<sup>i</sup>* <sup>−</sup> *<sup>K</sup>*<sup>2</sup> *k*2 *i*

The fundamental equation with the E-L formulation is given by [1, 2]

*<sup>ξ</sup>iD<sup>i</sup>* <sup>=</sup> *<sup>K</sup>*

*<sup>D</sup><sup>i</sup>* <sup>=</sup> <sup>D</sup>(0) *<sup>i</sup>* **e** (0) *<sup>i</sup>* <sup>+</sup> <sup>D</sup>(1) *<sup>i</sup>* **e** (1) *<sup>i</sup>* .

**2.1. Description of the** *n***-beam Ewald-Laue dynamical diffraction theory**

*<sup>D</sup><sup>i</sup>* <sup>=</sup> ∑*χhi*−*hj*

Here, *ki* is wavenumber of the *i*th numbered Bloch wave whose wave vector is **k**<sup>0</sup> + **h***<sup>i</sup>* where **k**<sup>0</sup> is the wave vector of the forward-diffracted wave in the crystal, *K*(= 1/*λ*) is the wavenumber of X-rays in vacuum, *D<sup>i</sup>* and *D<sup>j</sup>* are complex amplitude vectors of the *i*th and *j*th numbered Bloch waves, ∑ is an infinite summation for all combinations of *i* and *j*, *<sup>χ</sup>hi*−*hj* is Fourier coefficient of electric susceptibility and [*Dj*]⊥**k***<sup>i</sup>* is component vector of *D<sup>j</sup>*

By applying an approximation that *ki* + *K* ≈ 2*ki* to (1), the following equation is obtained,

<sup>2</sup> ∑*χhi*−*hj*

Let the electric displacement vector *D<sup>i</sup>* be represented by a linear combination of scalar

 *Dj* ⊥**k***<sup>i</sup>*

where *ξ<sup>i</sup>* = *ki* − *K*.

 *Dj* ⊥**k***<sup>i</sup>*

. (1)

, (2)

be published elsewhere.

perpendicular to **k***i*, respectively.

amplitudes as follows:

$$\mathbf{e}\_{j}^{(m)} = \mathbf{S}\_{i,j}^{(m)} \mathbf{s}\_{i} + \mathbf{C}\_{i,j}^{(0,m)} \mathbf{e}\_{i}^{(0)} + \mathbf{C}\_{i,j}^{(1,m)} \mathbf{e}\_{i}^{(1)} \,\prime \tag{4}$$

where *i* and *j* are ordinal numbers of waves (*i*, *j* ∈ {0, 1, 2, ··· , *n* − 1}) and *l* and *m* are ordinal numbers of polarization state (*l*, *m* ∈ {0, 1}). When deriving (3) from (2), all reciprocal lattice nodes lying on the surface of Ewald sphere are assumed to be on a circle in reciprocal space. Then number of waves *n* are limited to be *n* ∈ {3, 4, 5, 6, 8, 12} even in the case of cubic crystals with the highest symmetry. <sup>Θ</sup>*<sup>B</sup>* is the angle spanned by −→*PQ* and **<sup>k</sup>***<sup>i</sup>* which is an identical value for every *i* (*i* ∈ {0, 1, 2, ··· , *n* − 1}), where *P* and *Q* are centers of the Ewald sphere and the circle on which the reciprocal lattice nodes lie, respectively. *ξ* is such a value that

$$
\overrightarrow{P\_1 P\_1^\flat} = -\mathfrak{F} \overrightarrow{P} \overrightarrow{Q} / \left| \overrightarrow{P} \overrightarrow{Q} \right|. \tag{5}
$$

where *P*� <sup>1</sup> is the common initial point of **k***<sup>i</sup>* [whose terminal points are *Hi* (*i* ∈ {0, 1, ··· , *n* − 1})] and *P*<sup>1</sup> is a point on the sphere whose distance from the origin *O* of reciprocal space is *K*. Hereafter, this surface of sphere is approximated as a plane whose distance from *O* is *K* in the vicinity of the Laue point *La* whose distance from *Hi* (*i* ∈ {0, 1, ··· , *n* − 1}) is the identical value *K*. For description in the next section, it is described here that −−→ *P*1*P*� <sup>1</sup> is represented by a linear combination of **s***i*, **e** (0) *<sup>i</sup>* and **e** (1) *<sup>i</sup>* as follows:

$$\overrightarrow{P\_1 P\_1^\dagger} = -\xi \left( \cos \Theta\_B \mathbf{s}\_i + \eta\_i^{(0)} \sin \Theta\_B \mathbf{e}\_i^{(0)} + \eta\_i^{(1)} \sin \Theta\_B \mathbf{e}\_i^{(1)} \right) \dots$$

*P*<sup>1</sup> is such a point that

$$
\overrightarrow{P\_1 L a} = K \left( \boldsymbol{\beta}^{(0)} \mathbf{e}\_0^{(0)} + \boldsymbol{\beta}^{(1)} \mathbf{e}\_0^{(1)} \right). \tag{6}
$$

(2) and (3) can also be represented using matrices and vector as follows:

$$
\xi \cos \Theta\_{\mathsf{B}} \mathsf{E} \mathcal{D} = \mathsf{A} \mathcal{D}.\tag{7}
$$

Here, **E** is a unit matrix of size 2*n*, *D* is a amplitude column vector of size 2*n* whose *q*th element is *D*(*m*) *<sup>j</sup>* (*q* = 2*j* + *m* + 1) and **A** is a square matrix of size 2*n* whose element *ap*,*<sup>q</sup>* is given by

$$a\_{p,q} = \frac{K}{2} \chi\_{h\_i - h\_j} \mathbf{C}\_{i,j}^{(l,m)} - \delta\_{p,q} K \left( \mathbf{S}\_{i,0}^{(0)} \boldsymbol{\mathcal{B}}^{(0)} + \mathbf{S}\_{i,0}^{(1)} \boldsymbol{\mathcal{B}}^{(1)} \right).$$

Here, *p* = 2*i* + *l* + 1 and *δp*,*<sup>q</sup>* is Kronecker delta. 2*n* couples of *ξ* and *D* can be obtained by solving eigenvalue-eigenvector problem of (7). This problem was solved by Colella [3] for the first time. Dispersion surfaces on which the initial point *P*� <sup>1</sup> of wave vectors of Bloch waves should be, is given as 2*n* sets of eigenvalues for (7).

#### **2.2. Derivation of the** *n***-beam Takagi-Taupin equation from the Ewald-Laue theory**

In this section, the *n*-beam theory of T-T formulation is derived by Fourier-transforming the *n*-beam E-L theory described by (3).

A general solution of dynamical diffraction theory is considered to be coherent superposition of Bloch plane-wave system when X-ray wave field **D**˜ (**r**) is given as follows:

$$\tilde{\mathbf{D}}(\mathbf{r}) = \sum\_{i=0}^{n-1} \sum\_{l=0}^{1} \mathbf{e}\_i^{(l)} D\_i^{(l)}(\mathbf{r}) \exp\left(-\mathrm{i}2\pi \overrightarrow{\mathrm{La}H\_i} \cdot \mathbf{r}\right),\tag{8}$$

where **r** is the location vector. For the following description, **r** is described by a linear combination of **s***i*, **e** (0) *<sup>i</sup>* and **e** (1) *<sup>i</sup>* as follows,

$$\mathbf{r} = s\_l \mathbf{s}\_l + e\_i^{(0)} \mathbf{e}\_i^{(0)} + e\_i^{(1)} \mathbf{e}\_i^{(1)}.\tag{9}$$

The amplitude of the *i*th component wave whose polarization state is *l* is described as,

$$\begin{split} \mathcal{D}\_{i}^{(l)}(\Delta \mathbf{k}) \exp\left(-\mathrm{i}2\pi \overrightarrow{P\_{1}^{\prime}H\_{i}} \cdot \mathbf{r}\right) &= \mathcal{D}\_{i}^{(l)}(\Delta \mathbf{k}) \exp\left(-\mathrm{i}2\pi \Delta \mathbf{k} \cdot \mathbf{r}\right) \exp\left(-\mathrm{i}2\pi \overrightarrow{LaH\_{i}} \cdot \mathbf{r}\right), \\ & \text{where } \Delta \mathbf{k} = \overrightarrow{P\_{1}^{\prime}La}. \end{split}$$

In this section, the amplitude of plane wave whose wave vector is **k***<sup>i</sup>* and polarization state is *<sup>l</sup>* is denoted by <sup>D</sup>(*l*) *<sup>i</sup>* (Δ**k**) in place of <sup>D</sup>(*l*) *<sup>i</sup>* in order to clarify this value depends on <sup>Δ</sup>**k**. *<sup>D</sup>*(*l*) *<sup>i</sup>* (**r**) in (8) is represented by superposing coherently <sup>D</sup>(*l*) *<sup>i</sup>* (Δ**k**) as follows:

$$D\_i^{(l)}(\mathbf{r}) = \int\_{\Delta \mathbf{k}}^{D.S.} \mathcal{D}\_i^{(l)}(\Delta \mathbf{k}) \exp\left(-\mathbf{i}2\pi\Delta \mathbf{k} \cdot \mathbf{r}\right) \mathrm{d}S\_k. \tag{10}$$

Substituting (4) with *j* = 0, (5), (6) and (9) into (10),

$$\begin{split} D\_{i}^{(l)}(\mathbf{r}) &= \int\_{\Delta \mathbf{k}}^{D.S.} \mathcal{D}\_{i}^{(l)}(\Delta \mathbf{k}) \\ &\times \exp\left\{-\mathrm{i}2\pi \left[ \left( \xi \cos \Theta\_{\mathsf{B}} + K\beta^{(0)} S\_{i,0}^{(0)} + K\beta^{(1)} S\_{i,0}^{(1)} \right) s\_{i} + T\_{i} (\xi\_{\prime} e\_{i}^{(0)}, e\_{i}^{(1)}) \right] \right\} \mathrm{d} \mathbf{S}\_{k}. \tag{11} \end{split}$$

Here, *<sup>D</sup>*.*S*. <sup>Δ</sup>*<sup>k</sup>* d*Sk* means an integration over the dispersion surfaces in reciprocal space and *Ti*(*ξ*,*e* (0) *<sup>i</sup>* ,*e* (1) *<sup>i</sup>* ) is a term that does not depend on *si*. *<sup>∂</sup>D*(*l*) *<sup>i</sup>* (**r**)/*∂si* can be calculated as follows:

$$\begin{split} \frac{\partial}{\partial \mathbf{s}\_{l}} D\_{i}^{(l)}(\mathbf{r}) &= \frac{\partial}{\partial \mathbf{s}\_{l}} \int\_{\Delta \mathbf{k}}^{D.S.} \mathcal{D}\_{i}^{(l)}(\Delta \mathbf{k}) \exp\left(-\mathrm{i}2\pi \Delta \mathbf{k} \cdot \mathbf{r}\right) \mathrm{d}\mathbf{S}\_{k} \\ &= \int\_{\Delta \mathbf{k}}^{D.S.} \frac{\partial}{\partial \mathbf{s}\_{l}} \left[\mathcal{D}\_{i}^{(l)}(\Delta \mathbf{k}) \exp\left(-\mathrm{i}2\pi \Delta \mathbf{k} \cdot \mathbf{r}\right)\right] \mathrm{d}\mathbf{S}\_{k} \\ &= -\mathrm{i}2\pi \int\_{\Delta \mathbf{k}}^{D.S.} \left[\mathcal{J} \cos \Theta\_{\mathcal{B}} + K \left(\mathcal{S}\_{i,0}^{(0)} \mathcal{J}^{(0)} + \mathcal{S}\_{i,0}^{(1)} \mathcal{J}^{(1)}\right)\right] \mathcal{D}\_{i}^{(l)}(\Delta \mathbf{k}) \exp\left(-\mathrm{i}2\pi \Delta \mathbf{k} \cdot \mathbf{r}\right) \mathrm{d}\mathbf{S}\_{k}. \end{split} \tag{12}$$

Substituting (3) into (12),

4 Will-be-set-by-IN-TECH

Here, **E** is a unit matrix of size 2*n*, *D* is a amplitude column vector of size 2*n* whose *q*th

Here, *p* = 2*i* + *l* + 1 and *δp*,*<sup>q</sup>* is Kronecker delta. 2*n* couples of *ξ* and *D* can be obtained by solving eigenvalue-eigenvector problem of (7). This problem was solved by Colella [3] for the

In this section, the *n*-beam theory of T-T formulation is derived by Fourier-transforming the

A general solution of dynamical diffraction theory is considered to be coherent superposition

*<sup>i</sup>* (**r**) exp

where **r** is the location vector. For the following description, **r** is described by a linear

where <sup>Δ</sup>**<sup>k</sup>** <sup>=</sup> −−→

In this section, the amplitude of plane wave whose wave vector is **k***<sup>i</sup>* and polarization state is

*S*(0)

*<sup>i</sup>*,0 <sup>+</sup> *<sup>K</sup>β*(1)

(0) *<sup>i</sup>* **e** (0) *<sup>i</sup>* + *e* (1) *<sup>i</sup>* **e** (1)

The amplitude of the *i*th component wave whose polarization state is *l* is described as,

 −i2*π*

*<sup>i</sup>* (Δ**k**) exp (−i2*π*Δ**k** · **r**) exp

*P*� <sup>1</sup>*La*. −−→*LaHi* · **<sup>r</sup>** 

**2.2. Derivation of the** *n***-beam Takagi-Taupin equation from the Ewald-Laue**

*<sup>j</sup>* (*q* = 2*j* + *m* + 1) and **A** is a square matrix of size 2*n* whose element *ap*,*<sup>q</sup>* is

*<sup>i</sup>*,0 *<sup>β</sup>*(0) <sup>+</sup> *<sup>S</sup>*(1)

*<sup>i</sup>*,0 *<sup>β</sup>*(1) .

<sup>1</sup> of wave vectors of Bloch waves

, (8)

−−→*LaHi* · **<sup>r</sup>** ,

*<sup>i</sup>* (**r**)

d*Sk*. (11)

*<sup>i</sup>* . (9)

 −i2*π*

*<sup>i</sup>* in order to clarify this value depends on <sup>Δ</sup>**k**. *<sup>D</sup>*(*l*)

*<sup>i</sup>* (Δ**k**) exp (−i2*π*Δ**k** · **r**) d*Sk*. (10)

*si* + *Ti*(*ξ*,*e*

(0) *<sup>i</sup>* ,*e* (1) *<sup>i</sup>* ) 

*<sup>i</sup>* (Δ**k**) as follows:

*S*(1) *i*,0 

 *S*(0)

element is *D*(*m*)

*ap*,*<sup>q</sup>* <sup>=</sup> *<sup>K</sup>*

should be, is given as 2*n* sets of eigenvalues for (7).

**D**˜ (**r**) =

(0) *<sup>i</sup>* and **e**

 −i2*π* *n*−1 ∑ *i*=0

(1)

−−→ *P*� <sup>1</sup>*Hi* · **r** 

*<sup>i</sup>* (Δ**k**) in place of <sup>D</sup>(*l*)

 *D*.*S*. Δ**k**

*ξ* cos Θ*<sup>B</sup>* + *Kβ*(0)

D(*l*)

in (8) is represented by superposing coherently <sup>D</sup>(*l*)

*D*(*l*) *<sup>i</sup>* (**r**) =

Substituting (4) with *j* = 0, (5), (6) and (9) into (10),

D(*l*) *<sup>i</sup>* (Δ**k**)

 *D*.*S*. Δ**k**

× exp −i2*π* 

*n*-beam E-L theory described by (3).

<sup>2</sup> *<sup>χ</sup>hi*−*hj*

first time. Dispersion surfaces on which the initial point *P*�

*C*(*l*,*m*) *<sup>i</sup>*,*<sup>j</sup>* − *δp*,*qK*

of Bloch plane-wave system when X-ray wave field **D**˜ (**r**) is given as follows:

1 ∑ *l*=0 **e** (*l*) *<sup>i</sup> <sup>D</sup>*(*l*)

*<sup>i</sup>* as follows,

**r** = *si***s***<sup>i</sup>* + *e*

<sup>=</sup> <sup>D</sup>(*l*)

given by

**theory**

combination of **s***i*, **e**

D(*l*)

*<sup>l</sup>* is denoted by <sup>D</sup>(*l*)

*D*(*l*) *<sup>i</sup>* (**r**) =

*<sup>i</sup>* (Δ**k**) exp

$$\frac{\partial}{\partial S\_i} D\_i^{(l)}(\mathbf{r}) = -i\pi K \int\_{\Delta \mathbf{k}}^{D.S.} \sum\_{j=0}^{n-1} \sum\_{m=0}^{1} \mathbb{C}\_{i,j}^{(l,m)} \chi\_{h\_i - h\_j} \mathcal{D}\_j^{(m)}(\Delta \mathbf{k}) \exp\left(-i2\pi \Delta \mathbf{k} \cdot \mathbf{r}\right) \mathrm{d}S\_k$$

$$= -i\pi K \sum\_{j=0}^{n-1} \sum\_{m=0}^{1} \mathbb{C}\_{i,j}^{(l,m)} \chi\_{h\_i - h\_j} \int\_{\Delta \mathbf{k}}^{D.S.} \mathcal{D}\_j^{(m)}(\Delta \mathbf{k}) \exp\left(-i2\pi \Delta \mathbf{k} \cdot \mathbf{r}\right) \mathrm{d}S\_k. \tag{13}$$

Incidentally, when the crystal is perfect, the electric susceptibility *χ*(**r**) is represented by Fourier series as *χ*(**r**) = ∑**h***<sup>i</sup> χhi* exp[−i2*π***h***<sup>i</sup>* · **r**]. However, when the crystal has a lattice displacement field of **u**(**r**), the electric susceptibility is approximately given by *χ*[**r** − **u**(**r**)] and represented by Fourier series as follows,

$$\chi[\mathbf{r} - \mathbf{u}(\mathbf{r})] = \sum\_{\mathbf{h}\_l} \chi\_{\mathbf{h}\_l} \exp[\mathbf{i}2\pi \mathbf{h}\_l \cdot \mathbf{u}(\mathbf{r})] \exp(-\mathbf{i}2\pi \mathbf{h}\_l \cdot \mathbf{r}).$$

Then, in the case of crystal with a lattice displacement field of **<sup>u</sup>**(**r**), *<sup>χ</sup>hi*−*hj* can be replaced by *<sup>χ</sup>hi*−*hj* exp[i2*π*(**h***<sup>i</sup>* − **<sup>h</sup>***j*) · **<sup>u</sup>**(**r**)]. Therefore, the following equation is obtained from (13),

$$\frac{\partial}{\partial \mathbf{s}\_i} D\_i^{(l)}(\mathbf{r}) = -i\pi K \sum\_{j=0}^{n-1} \sum\_{m=0}^{1} \mathbb{C}\_{i,j}^{(l,m)} \chi\_{h\_i - h\_j} \exp\left[i2\pi (\mathbf{h}\_i - \mathbf{h}\_j) \cdot \mathbf{u}(\mathbf{r})\right] D\_j^{(m)}(\mathbf{r}), \tag{14}$$
 
$$\text{where, } \quad i, j \in \{0, 1, \dots, n-1\}, n \in \{3, 4, 5, 6, 8, 12\},$$
 
$$l, m \in \{0, 1\}.$$

The above equation is nothing but the *n*-beam T-T equation that appeared as eq. (4) in reference [17].

#### **2.3. Derivation of the** *n***-beam E-L dynamical theory from the T-T equation**

In this section, it is described that the *n*-beam E-L theory given by (3) can be derived from the *n*-beam T-T equation (14).

#### 6 Will-be-set-by-IN-TECH 72 Recent Advances in Crystallography

When plane-wave X-rays are incident on the crystal to excite 2*n* tie points on the dispersion surfaces, each Bloch plane-wave system is described by

$$\tilde{\mathcal{D}} = \sum\_{i=0}^{n-1} \sum\_{l=0}^{1} \mathbf{e}\_i^{(l)} \mathcal{D}\_i^{(l)} \exp\left(-\mathbf{i}2\pi\Delta\mathbf{k}\cdot\mathbf{r}\right) \exp\left(-\mathbf{i}2\pi\overline{\mathbf{L}aH\_l}\cdot\mathbf{r}\right).$$

Even when *D*(*l*) *<sup>i</sup>* (**r**) = <sup>D</sup>(*l*) *<sup>i</sup>* exp(−i2*π***k**(*l*) *<sup>i</sup>* · **<sup>r</sup>**), *<sup>D</sup>*(*l*) *<sup>i</sup>* (**r**) should satisfy (14) with **u**(**r**) = 0,

$$\frac{\partial}{\partial \mathbf{s}\_i} \left[ \mathcal{D}\_i^{(l)} \exp \left( -\mathbf{i} 2\pi \Delta \mathbf{k} \cdot \mathbf{r} \right) \right] = -\mathbf{i} \pi \mathbf{K} \sum\_{j=0}^{n-1} \sum\_{m=0}^{1} \mathbb{C}\_{i,j}^{(l,m)} \chi\_{h\_i - h\_j} \left[ \mathcal{D}\_j^{(m)} \exp \left( -\mathbf{i} 2\pi \Delta \mathbf{k} \cdot \mathbf{r} \right) \right]. \tag{15}$$

Applying the same procedure as used when deriving (11),

$$\begin{split} &\frac{\partial}{\partial s\_{i}}\left[\mathcal{D}\_{i}^{(l)}\exp\left(-\mathrm{i}2\pi\Delta\mathbf{k}\cdot\mathbf{r}\right)\right] \\ &=\mathcal{D}\_{i}^{(l)}\frac{\partial}{\partial s\_{i}}\exp\left\{-\mathrm{i}2\pi\left[\left(\xi\cos\Theta\_{B}+\mathsf{K}\beta^{(0)}S\_{i,0}^{(0)}+\mathsf{K}\beta^{(1)}S\_{i,0}^{(1)}\right)s\_{i}+T\_{i}(\xi,\mathcal{E}\_{i}^{(0)},\mathcal{E}\_{i}^{(1)})\right]\right\} \\ &=-\mathrm{i}2\pi\left(\xi\cos\Theta\_{B}+\mathsf{K}\beta^{(0)}S\_{i,0}^{(0)}+\mathsf{K}\beta^{(1)}S\_{i,0}^{(1)}\right)\left[\mathcal{D}\_{i}^{(l)}\exp\left(-\mathrm{i}2\pi\Delta\mathbf{k}\cdot\mathbf{r}\right)\right]. \end{split} \tag{16}$$

Comparing (15) and (16), the same equation as (3) is obtained. The equivalence between the *n*-beam E-L and T-T X-ray dynamical diffraction theories (*n* ∈ {3, 4, 5, 6, 8, 12}) described by a Fourier transform as defined by (10) is verified. As far as the present authors know, description on this equivalence between the E-L and T-T dynamical diffraction theories for two-beam case is found just in section 11.3 of Authier's book [1].

#### **3. Algorithm to solve the theory**

Figure 1(*a*) and 1(*b*) are schematic drawings for explanation of the algorithm to solve the *n*-beam T-T equation (14) for a six-beam case whose computer-simulated and experimentally obtained results are shown in Figure 9 of the present chapter. Vectors −−−−−→ *R*(0) *<sup>i</sup> <sup>R</sup>*(1) in Figure 1(*a*) are parallel to **s***i*. When the length of −−−−−→ *R*(0) *<sup>i</sup> <sup>R</sup>*(1) is sufficiently small compared with the extinction length −1/(*χ*0*K*) of the forward diffraction, The T-T equation (14) can be approximated by

$$\frac{D\_i^{(l)}(\mathcal{R}^{(1)}) - D\_i^{(l)}(\mathcal{R}\_i^{(0)})}{\left| \overline{R\_i^{(0)} \mathcal{R}^{(1)}} \right|}$$

$$= -\text{i}\pi K \sum\_{j=0}^{n-1} \sum\_{m=0}^1 \left\{ \chi\_{h\_i - h\_j} \exp\left[ \text{i}2\pi \left( \mathbf{h}\_i - \mathbf{h}\_j \right) \cdot \mathbf{u}(Rm\_i) \right] \right. $$

$$\times C\_{i,j}^{(l,m)} \left[ D\_j^{(m)}(\mathcal{R}\_i^{(0)}) + D\_j^{(m)}(\mathcal{R}^{(1)}) \right] / 2 \right\}. \tag{17}$$

72 Recent Advances in Crystallography X-ray N-beam Takagi-Taupin Dynamical Theory and N-beam Pinhole Topographs Experimentally Obtained and Computer-Simulated <sup>7</sup> 73 X-Ray N-Beam Takagi-Taupin Dynamical Theory and N-Beam Pinhole Topographs Experimentally Obtained and Computer-Simulated

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When plane-wave X-rays are incident on the crystal to excite 2*n* tie points on the dispersion

*<sup>i</sup>* exp (−i2*π*Δ**k** · **r**) exp

1 ∑ *m*=0

*C*(*l*,*m*) *<sup>i</sup>*,*<sup>j</sup> χhi*−*hj*

*S*(0)

*S*(1) *i*,0 D(*l*)

Comparing (15) and (16), the same equation as (3) is obtained. The equivalence between the *n*-beam E-L and T-T X-ray dynamical diffraction theories (*n* ∈ {3, 4, 5, 6, 8, 12}) described by a Fourier transform as defined by (10) is verified. As far as the present authors know, description on this equivalence between the E-L and T-T dynamical diffraction theories for

Figure 1(*a*) and 1(*b*) are schematic drawings for explanation of the algorithm to solve the *n*-beam T-T equation (14) for a six-beam case whose computer-simulated and experimentally

the extinction length −1/(*χ*0*K*) of the forward diffraction, The T-T equation (14) can be

−−−−−→ *R*(0)

*<sup>i</sup>*,0 <sup>+</sup> *<sup>K</sup>β*(1)

*<sup>i</sup>* · **<sup>r</sup>**), *<sup>D</sup>*(*l*)

*n*−1 ∑ *j*=0 <sup>−</sup>i2*π*−−→*LaHi* · **<sup>r</sup>**

*<sup>i</sup>* (**r**) should satisfy (14) with **u**(**r**) = 0,

 D(*m*)

*S*(1) *i*,0 

*<sup>i</sup>* exp (−i2*π*Δ**k** · **r**)

 .

*<sup>j</sup>* exp (−i2*π*Δ**k** · **r**)

*si* + *Ti*(*ξ*,*e*

(0) *<sup>i</sup>* ,*e* (1) *<sup>i</sup>* ) 

−−−−−→ *R*(0)

*<sup>i</sup> <sup>R</sup>*(1) is sufficiently small compared with

· **u**(*Rmi*)

. (17)

*<sup>i</sup> <sup>R</sup>*(1) in Figure

 . (15)

. (16)

surfaces, each Bloch plane-wave system is described by

1 ∑ *l*=0 **e** (*l*) *<sup>i</sup>* <sup>D</sup>(*l*)

*<sup>i</sup>* exp(−i2*π***k**(*l*)

Applying the same procedure as used when deriving (11),

= −i*πK*

*ξ* cos Θ*<sup>B</sup>* + *Kβ*(0)

*<sup>i</sup>*,0 <sup>+</sup> *<sup>K</sup>β*(1)

*S*(0)

two-beam case is found just in section 11.3 of Authier's book [1].

obtained results are shown in Figure 9 of the present chapter. Vectors

*<sup>i</sup>* (*R*(0) *<sup>i</sup>* )

*n*−1 ∑ *i*=0

*<sup>D</sup>*˜ <sup>=</sup>

*<sup>i</sup>* (**r**) = <sup>D</sup>(*l*)

*<sup>i</sup>* exp (−i2*π*Δ**k** · **r**)

*<sup>i</sup>* exp (−i2*π*Δ**k** · **r**)

*ξ* cos Θ*<sup>B</sup>* + *Kβ*(0)

Even when *D*(*l*)

*∂ ∂si* D(*l*)

<sup>=</sup> <sup>D</sup>(*l*) *i ∂ ∂si* exp −i2*π* 

= −i2*π*

approximated by

**3. Algorithm to solve the theory**

1(*a*) are parallel to **s***i*. When the length of

*D*(*l*)

*<sup>i</sup>* (*R*(1)) <sup>−</sup> *<sup>D</sup>*(*l*)

−−−−−→ *R*(0) *<sup>i</sup> <sup>R</sup>*(1) 

> *n*−1 ∑ *j*=0

> > *D*(*m*) *<sup>j</sup>* (*R*(0)

1 ∑ *m*=0

*<sup>χ</sup>hi*−*hj* exp

*<sup>i</sup>* ) + *<sup>D</sup>*(*m*)

 i2*π* **h***<sup>i</sup>* − **h***<sup>j</sup>* 

*<sup>j</sup>* (*R*(1) ) /2 

 

×*C*(*l*,*m*) *i*,*j*

= −i*πK*

*∂ ∂si* D(*l*)

**Figure 1.** This Figure shows small hexagonal pyramids used when solving the T-T equation (14) in a six-beam case whose results are shown in Figure 9.

The above equation (17) can be described using matrix and vectors as follows:

$$\begin{split} \mathbf{A} &= \mathbf{B} \mathbf{D} \\ \text{where, } a\_p &= -\mathbf{i}\frac{1}{2}\pi K \sum\_{j=0}^{n-1} \sum\_{m=0}^{1} \chi\_{h\_i - h\_j} \exp\left[i2\pi (\mathbf{h}\_i - \mathbf{h}\_j) \cdot \mathbf{u}(\mathcal{R}m\_i)\right] \mathbf{C}\_{i,j}^{(l,m)} \mathbf{D}\_j^{(m)}(\mathbf{R}\_i^{(0)}) \\ &+ \frac{\mathbf{D}\_i^{(l)}(\mathbf{R}\_i^{(0)})}{\left|\mathbf{R}\_i^{(0)}\mathbf{R}^{(1)}\right|}, \\ b\_{p,q} &= \mathbf{i}\frac{1}{2}\pi K \chi\_{h\_l - h\_j} \exp\left[i2\pi (\mathbf{h}\_i - \mathbf{h}\_j) \cdot \mathbf{u}(\mathcal{R}m\_i)\right] \mathbf{C}\_{i,j}^{(l,m)} + \frac{\delta\_{p,q}}{\left|\mathbf{R}\_i^{(0)}\mathbf{R}^{(1)}\right|}, \\ d\_q &= D\_j^{(m)}(\mathbf{R}^{(1)}), \\ p &= 2i + l + 1, \\ q &= 2j + m + 1. \end{split} \tag{18}$$

#### 8 Will-be-set-by-IN-TECH 74 Recent Advances in Crystallography

Here, **A** and **D** are column vectors of size 2*n* whose *p*th and *q*th elements are *ap* and *dq*, respectively, and **B** is a square matrix of size 2*n* whose element of the *p*th column and the *q*th raw is *bp*,*q*.

**Figure 2.** This Figure shows a top view of Figure 1(*b*).

Figure 2 is a top view of Figure 1(*b*). The X-ray amplitudes *<sup>D</sup>*(*m*) *<sup>j</sup>* (*R*(1) *<sup>i</sup>* ) were calculated from the X-ray amplitudes at the incidence point *D*(*l*) <sup>0</sup> (*Rinc*) of the crystal surface. In this case, 0 0 0-forward-diffracted and 4 0 4-, 4 2 6-, 0 6 6-, 2 6 4- and 2 2 0-reflected X-rays are simultaneously strong. The angle spanned by *nx*- and *ny*-axes is 120◦. −−−−−→ *RincR*(1) *<sup>i</sup>* in Figure 1(*b*) are parallel to the wave vectors of 0 0 0-forward-diffracted and 4 0 4-, 4 2 6-, 0 6 6-, 2 6 4 and 2 2 0-reflected X-rays. As a boundary condition on the crystal surface, amplitude array *Deven*(*i*,*l*,*nx*,*ny*) has nonzero value (unity) when (*i*,*l*,*nx*,*ny*)=(0,0,0,0) or (*i*,*l*,*nx*,*ny*)=(0,1,0,0). On the first layer, nonzero X-ray amplitudes *Dodd*(*j*, *m*, *nx*, *ny*) are calculated when (*nx*, *ny*) = [*n*� *<sup>x</sup>*(*i*), *n*� *<sup>y</sup>*(*i*)] (*i* ∈ {0, 1, ··· , *n* − 1}). Here, [*n*� *<sup>x</sup>*(*i*), *n*� *<sup>y</sup>*(*i*)] = (0, 0),(0, 2),(1, 3),(3, 3),(3, 2) and (1, 0) for *i* = 0, 1, 2, 3, 4, 5, respectively. In general, *Deven*(*i*,*l*,*nx*,*ny*) [or *Dodd*(*i*,*l*,*nx*,*ny*)] is calculated as *D*(*l*) *<sup>i</sup>* (*R*(1)) by substituting *Dodd*(*j*,*m*,*nx* <sup>−</sup> *<sup>n</sup>*� *<sup>x</sup>*(*i*),*ny* − *n*� *<sup>y</sup>*(*i*)) [or *Deven*(*j*,*m*,*nx* − *n*� *<sup>x</sup>*(*i*),*ny* − *n*� *<sup>y</sup>*(*i*))] into *<sup>D</sup>*(*m*) *<sup>j</sup>* (*R*(0) *<sup>i</sup>* ) in (17). The calculation was performed layer by layer scanning *nx* and *ny* in a range of *NMin*[*n*� *<sup>x</sup>*(*i*)] ≤ *nx* ≤ *N Max*[*n*� *<sup>x</sup>*(*i*)] and *NMin*[*n*� *<sup>y</sup>*(*i*)] ≤ *ny* ≤ *N Max*[*n*� *<sup>y</sup>*(*i*)], where *<sup>N</sup>* is the ordinal number of layer. The values of *<sup>χ</sup>hi*−*hj* were calculated by using *XOP* version 2.3 [6].

#### **4. Experimental**

#### **4.1. Phase-retarder system**

When taking four-, five-, six- and eight-beam pinhole topographs shown in section 5, the horizontally polarized synchrotron X-rays monochromated to be 18.245 keV with a water-cooled diamond monochromator system at BL09XU of SPring-8 were incident on the 'rotating four-quadrant phase retarder system' [15, 17].

Figure 3 shows (*a*) a schematic drawing of the phase retarder system and (*b*) a photograph of it. [100]-oriented four diamond crystals *PRn* (*n* ∈ {1, 2, 3, 4}) with thicknesses of 1.545, 74 Recent Advances in Crystallography X-ray N-beam Takagi-Taupin Dynamical Theory and N-beam Pinhole Topographs Experimentally Obtained and Computer-Simulated <sup>9</sup> 75 X-Ray N-Beam Takagi-Taupin Dynamical Theory and N-Beam Pinhole Topographs Experimentally Obtained and Computer-Simulated

8 Will-be-set-by-IN-TECH

Here, **A** and **D** are column vectors of size 2*n* whose *p*th and *q*th elements are *ap* and *dq*, respectively, and **B** is a square matrix of size 2*n* whose element of the *p*th column and the *q*th

case, 0 0 0-forward-diffracted and 4 0 4-, 4 2 6-, 0 6 6-, 2 6 4- and 2 2 0-reflected X-rays are

are parallel to the wave vectors of 0 0 0-forward-diffracted and 4 0 4-, 4 2 6-, 0 6 6-, 2 6 4 and 2 2 0-reflected X-rays. As a boundary condition on the crystal surface, amplitude array *Deven*(*i*,*l*,*nx*,*ny*) has nonzero value (unity) when (*i*,*l*,*nx*,*ny*)=(0,0,0,0) or (*i*,*l*,*nx*,*ny*)=(0,1,0,0). On the first layer, nonzero X-ray amplitudes *Dodd*(*j*, *m*, *nx*, *ny*) are calculated when (*nx*, *ny*) =

(1, 0) for *i* = 0, 1, 2, 3, 4, 5, respectively. In general, *Deven*(*i*,*l*,*nx*,*ny*) [or *Dodd*(*i*,*l*,*nx*,*ny*)] is

When taking four-, five-, six- and eight-beam pinhole topographs shown in section 5, the horizontally polarized synchrotron X-rays monochromated to be 18.245 keV with a water-cooled diamond monochromator system at BL09XU of SPring-8 were incident on the

Figure 3 shows (*a*) a schematic drawing of the phase retarder system and (*b*) a photograph of it. [100]-oriented four diamond crystals *PRn* (*n* ∈ {1, 2, 3, 4}) with thicknesses of 1.545,

*<sup>x</sup>*(*i*), *n*�

*<sup>x</sup>*(*i*)] ≤ *nx* ≤ *N Max*[*n*�

*<sup>y</sup>*(*i*)], where *<sup>N</sup>* is the ordinal number of layer. The values of *<sup>χ</sup>hi*−*hj* were calculated

*<sup>j</sup>* (*R*(1)

<sup>0</sup> (*Rinc*) of the crystal surface. In this

*<sup>y</sup>*(*i*)] = (0, 0),(0, 2),(1, 3),(3, 3),(3, 2) and

*<sup>x</sup>*(*i*)] and *NMin*[*n*�

*<sup>x</sup>*(*i*),*ny* − *n*�

*<sup>i</sup>* ) in (17). The calculation was performed layer by layer

−−−−−→ *RincR*(1)

*<sup>i</sup>* ) were calculated

*<sup>i</sup>* in Figure 1(*b*)

*<sup>y</sup>*(*i*)) [or *Deven*(*j*,*m*,*nx* −

*<sup>y</sup>*(*i*)] ≤ *ny* ≤

raw is *bp*,*q*.

[*n*� *<sup>x</sup>*(*i*), *n*�

*n*�

calculated as *D*(*l*)

**4. Experimental**

*<sup>x</sup>*(*i*),*ny* − *n*�

*N Max*[*n*�

**Figure 2.** This Figure shows a top view of Figure 1(*b*).

from the X-ray amplitudes at the incidence point *D*(*l*)

*<sup>y</sup>*(*i*)] (*i* ∈ {0, 1, ··· , *n* − 1}). Here, [*n*�

'rotating four-quadrant phase retarder system' [15, 17].

*<sup>j</sup>* (*R*(0)

*<sup>y</sup>*(*i*))] into *<sup>D</sup>*(*m*)

scanning *nx* and *ny* in a range of *NMin*[*n*�

by using *XOP* version 2.3 [6].

**4.1. Phase-retarder system**

Figure 2 is a top view of Figure 1(*b*). The X-ray amplitudes *<sup>D</sup>*(*m*)

simultaneously strong. The angle spanned by *nx*- and *ny*-axes is 120◦.

*<sup>i</sup>* (*R*(1)) by substituting *Dodd*(*j*,*m*,*nx* <sup>−</sup> *<sup>n</sup>*�

**Figure 3.** (*a*) is a schematic drawing of the 'rotating four-quadrant phase retarder system' (reproduction of Figure 3 in reference [17]). (*b*) is a photograph of it.

2.198, 1.565 and 2.633 mm were mounted on tangential-bar type goniometers such that the deviation angles Δ*θPRn* from the exact Bragg condition of 1 1 1 reflection in an asymmetric Laue geometry can be controlled. See Figure 4 in reference [17] for more detail. The four tangential-bar type goniometers were mounted in a *χ*-circle goniometer [see Figure 3 (*b*)] such that the whole system of the phase retarders can be rotated around the beam axis of transmitted X-rays. The rotation angle of the *χ*-circle *χPR* and Δ*θPRn* were controlled as summarized in Table 3 in reference [17] such that horizontal-linearly (LH), vertical-linearly (LV), right-screwed circularly (CR), left-screwed circularly (CL), −45◦-inclined-linearly (L−45) and +45◦-inclined-linearly (L+45) polarized X-rays were generated to be incident on the sample crystal.

In the cases of three- and twelve-beam pinhole topographs, horizontally polarized synchrotron X-rays monochromated to be 18.245 keV and to be 22.0 keV, respectively, but not transmitted through the phase retarder system were incident on the sample crystals.

### **4.2. Sample crystal**

Figure 4 is a reproduction of Fig 7 in reference [17] showing the experimental setup when the six-beam pinhole topographs shown in reference [17] were taken. Also in the case of *n*-beam pinhole topographs (*n* ∈ {3, 4, 5, 6, 8, 12}) the [1 1 1]-oriented floating-zone silicon crystal with thickness 9.6 mm (for three-, four-, five-, six- and eight-beam topographs) and 10.0 mm (for twelve-beam topographs) were also mounted on the four-axis goniometer whose *χ*-, *φ*-, *ω*and *θ*-axes can be rotated. Transmitted X-rays through the sample and two reflected X-rays were searched by three *PIN* photo diodes as shown in Figure 4. The positions of the two *PIN* photo diodes for detecting the reflected X-rays were determined using a laser beam guide reflected by a mirror. The mirror was mounted at the sample position on the goniometer whose angular positions were calculated such that the mirror reflects the laser beam to the direction of X-rays to be reflected by the sample crystal.

**Figure 4.** A schematic drawing of the goniometer on which the sample crystal was mount (reproduction of Figure 7 in reference [17]).

After adjusting the angular position of the goniometer such that the *n*-beam simultaneous reflection condition was satisfied, the size of slit *S* in Figure 3 (*a*) was limited to be 25×25 *μ*m.

*N* images of *n*-beam pinhole topographs were simultaneously recorded on an imaging plate placed behind the sample crystal.
